Dynamics Lecture3

8/13/2019 Dynamics Lecture3

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Structural Dynamics

Lecture 3

Outline of Lecture 3

Single-Degree-of-Freedom Systems (cont.)

Forced Vibrations due to Arbitrary Excitation.

DAlemberts Principle.

Vibrations due to Movable Support.

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art qua e xcitations.

Vibrations due to Indirectly Acting Dynamic Loads.


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Single-Degreee-of-Freedom Systems (cont.)

Forced Vibrations due to Arbitrary Excitation

Equations of motion of a linear viscous damped system:

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Box 1 : Equation of momentum

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: Impulse (momentum) of external and internal forces, [kg s].

Equation of momentum:

The increase of momentum of the particle during a time

interval is equal to the impulse, i.e. the time integral over the same

interval of all external and internal forces.


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The load is constant for and zero out-side this

interval. The load is applied at the time , where the displacement is

and the velocity is . The impulse of is equal to for

arbitrary . Assume that is small compared to the undamped

eigenperiod of the oscillator. Then, we may let without any

physical consequences for the response. The shape of the load is

arbitrary, only its impulse is important.


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is unaffected by the dynamic load and is continuous for due

to the inertia of the mass. This implies that the displacement at the

time immediately after the cease of the load is given as:

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velocity is searched immediately after the load has ceased.

Due to (3) it follows that:


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Then, for an impulsive dynamic load (short duration, large intensity), it

follows from (2):

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An impulsive load for applied at can be described by Diracs

delta function :

Especially, is denoted a unit impulse, if .


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Let a unit impulse be applied at to a SDOF oscillator at rest at

, i.e.

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The load history is divided into differential equivalent impulses of the

magnitude (hatched area). The response at the time from a

differential impulse applied at the time to a system at rest becomes:


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The response from all differential impulses in the time interval follows

from the superposition principle:

(12) is a particular integral for (1), known as Duhamels integral. (11)

describes the motion, if the impulse is applied at a system at rest at the

[0, t]

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time , i.e. . Hence, (12) describes the totalresponse of a system at rest at , i.e. .

Then, the total solution of (1) follows from (12) and the complementary

solution (eigenvibration) from the initial values and ,

cf. Lecture 2, Eq. (6):


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Example 1 : Undamped SDOF exposed to a ramp load

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. Then, in Eq. (9) reduces to


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The integrals in Eq. (15) may be evaluated analytically.


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Relation between and :

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The impulse response function and the frequency response function

describes the same dynamic system exposed to standardizeddynamic loads. There must be a relation between these two system

defining functions.


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Then, is determined by the inverse Fourier integral transform

Hence, and are mutual Fourier transforms.


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Box 2 : Formal derivation of Fouriers integral theorem

The complex representation of the Fourier series of

reads, cf. Lecture 1, Eqs. (21), (22):

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where for is given as


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for follows from .

(18) and (19) may be written as

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where


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As , so is described by a continuous frequency

parameter , (20) attains the form

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(22) merely represents the result of a formal mathematical operation, i.e.

the convergence of (20) towards (22) is not guaranteed. However, the

convergence can be proven, if is piece-wise differential and absolute

integrable, i.e. if


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DAlemberts Principle

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Newtons 2nd law of motion for the free mass reads:

The equation may be rewritten in the form:


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Formally, (26) may be interpreted as a statical equilibrium equation, where

a so-called inertial force is applied in the same direction as

(and ). Its magnitude is .

dAlemberts principle:

Inertial forces (inertial moments for rotational degrees of

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damping and spring forces in the direction of the degrees of freedom, andthe equations of motion are next obtained from static equilibrium

equations of the system.

dAlemberts principle is a formal operation easing the formulation of

dynamic equations of motion as an alternative to Newtons 2nd law ofmotion and to analytical dynamics. Inertial forces are fictitious quantities

with no physical meaning.


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Example 2 : Horizontal plane beam

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: Bending stiffness, [N m2].

: Mass per unit length, [kg/m].

SDOF system: The degree of freedom is selected as the verticaldisplacement of the mass. The beam is cut free from the damper andthe spring. The internal damping force and spring force areapplied as external forces with the signs as shown on Fig. 7b.


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Use of the equation of moment of momentum:

The equation of moment of momentum is formulated around point in

the clock-wise direction, cf. Lecture 1, Eq. (46):

Moment of external forces around (no contribution from the unknown

-

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Use of dAlemberts principle:

The inertial force is applied as shown on Fig. 7b. Statical moment

equilibrium around provides:

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The reaction is obtained from the vertical static force equilibrium

equation:


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The undamped angular eigenfrequency and the damping ratio become:

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Vibrations due to Movable Support

Horizontal support motion due to earthquakes or heavy traffic.

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Plane single-degree-of-freedom shear frame exposed to a horizontal

ground surface motion.

: Mass of infinite rigid storey beam, [kg]. : Total shear stiffness of both massless columns, [N/m].

: Linear viscous damping coefficient, [N s/m].

: Horizontal displacement of storey beam from static equilibrium

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, .

: Horizontal ground surface motion, [m]. : Shear force in both columns incl. the damper force, [N].

.

: Reaction force on the ground surface, [N]. .

The ground surface motion and its time derivatives and areassumed to be known as a function of time. may be measured

with an accelerometer or a seismograph.


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The storey mass is cut free from the columns and the damper, and the

shear force is applied on the free mass with the sign shown on Fig.

8b. Newtons 2nd law of motion provides:

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The motion of the storey beam relative to the ground surface becomes:


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Equation of motion and reaction force on the ground surface formulated in

the relative motion:

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(36) is preferred in earthquake engineering, because is directlyavailable, via measurements.


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Example 3 : Harmonic motion of vibration isolated mass

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: Vertical motion of mass from statical equilibrium state, [m].

: Vertical motion of ground surface from referential position, [m].


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Equation of motion of free mass:

Let be harmonically varying with the amplitude , the angular

frequency and the phase . Then, cf. Lecture 2, Eq. (33):

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The right-hand side of (38) becomes:


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Then, the stationary response follows from, see Lecture 2, Eqs. (38), (39):

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The dynamic amplification factor becomes:


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Earthquake Excitations

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(36) is rewritten on the form

represents the ground surface acceleration caused by a given design

earthground. From (47) the maximum relative displacement registered

during the earthquake may be obtained by numerical time integration:

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is denoted the spectral displacement. The spectral velocity and the

spectral acceleration are defined based on the spectral displacement as:


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Fig. 9b shows the variation of the normalized spectral velocity for the NS

component of the El Centro earthquake as a function of the damping ratio

and the undamped angular eigenfrequency of the structure. Noticethat is usually different from the maximum relative

acceleration, i.e.

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S l D i


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In earthquake resistant design is applied as a horizontal

inertial load on the structure in agreement with dAlemberts principle,

and next the structure is designed by usual static analysis. For the SDOFshear frame in Fig. 8 this provides the maximum relative displacement:

St t l D i


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Vibrations due to Indirectly Acting Dynamic Loads

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Often the dynamic load is acting indirectly on the masses via a transmissionthrough a linear elastic massless structure.

: Displacement of mass. Degree of freedom of the system.

: Displacement of the structure in the direction of the dynamicforce. Auxiliary degree of freedom.

: Flexibility coefficient. Displacement (rotation) of the degree of

freedom from a unit force (moment) applied at the degree of

freedom .

St t l D i


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DAlemberts principle is used. The displacement is caused by the

quasi-state contribution from the inertial load at the degree of

freedom and the external dynamic load at the degree of freedom. The linearity of the system (the superposition principle) provides:

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(53) is merely an application of the force methodof statics.

Structural Dynamics


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Box 3 : Flexibility coefficients

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: Bending stiffness.

: Length of structure.

: Moment distribution from unit force (moment) at the degree

of freedom .

Principle of complementary virtual work:

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From the right-hand side of (54) follows:

(55) is known as Maxwell-Bettis reciprocal theorem.

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If the degrees of freedom and represents a translation and a rotation,respectively, the theorem states that the displacement in the degree of

freedom from a unit moment acting in the degree of freedom is equalto the rotation in the degree of freedom from a unit force acting in thedegree of freedom .

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Summary of Lecture 3

Forced Vibration due to Arbitrary Excitation

Impulse of loads on a free particle. Equation of momentum.

Impulsive loads (short duration, large intensity). The displacement is

continuous, whereas the velocity is discontinuous.

Impulse response function . Related to the frequency response by

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Fouriers integral theorem.

Duhamels integral. Defines a particular integral (stationary motion) to theequation of motion.

DAlemberts Principle

Inertial force (moment) is applied to the free particle along with the

internal ( ) and external forces ( ). Dynamic equations ofmotion are formally obtained by static equilibrium equations.

The principle facilitates the formulation of equations of motion. Inertial

forces are fictitious quantities. The physical reality is Newtons 2nd law of

motion.

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Earthquake Excitations

Spectral displacement, .

Spectral acceleration, .

The spectral acceleration is used as an equivalent static load in earthquake

resistant structural design in accordance with dAlemberts principle.

Vibrations due to Indirectl Actin D namic Loads

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Dynamic load are not acting at the mass, but is transferred via a linear elasticmassless structure.

Flexibility coefficients (Maxwell-Bettis reciprocal theorem).

Dynamics Lecture3

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